Relationships between t distributions and normal distributions

Question

In Gelman's Bayesian Data Analysis:

The t distribution is the marginal posterior distribution for the normal mean with unknown variance and conjugate prior distribution and can be interpreted as a mixture of normals with common mean and variances that follow an inverse-gamma distribution.

• What precisely (in terms of mathematical formulas) does it mean by "the t distribution is the marginal posterior distribution for the normal mean with unknown variance"?

• In the table on Wikipedia, I didn't find that the t distribution is a conjugate prior distribution for a normal distribution. So where does "conjugate prior distribution" come from?

Answer 1

I'll let someone else address your first question, but regarding the second bullet point, @Glen_b is right: the $t$ distribution is the posterior, not the prior. The problem here is that there is an ambiguity in the text concerning what phrases are being joined by the "and". Consider these two possibilities:

1. The t distribution is $($the marginal posterior distribution for the normal mean with unknown variance$)$ and [it is also the] conjugate prior distribution...

   vs

2. The t distribution is the marginal posterior distribution for the normal mean $($with unknown variance and [when the] conjugate prior distribution [had been used]$)$...

The correct interpretation is #2.

Answer 2

Getting away from the heavy duty math and talking in plain English, I just think of the t distribution as a Normal distribution adjusted for greater uncertainty associated with small samples (low DFs in the table). As a result, the t distribution has wider or fatter tails than the Normal Distribution resulting in much wider Confidence Intervals (reflecting the greater uncertainty with small samples). Once n ~ 60 or more, the two distributions become undistinguishable. And, if you conduct hypothesis testing with either the t distribution or the Normal distribution you will get essentially the same p values.